We have investigated the scattering properties of an individual core-shell nanoparticle using the Mie theory, which can be tuned to support both electric and magnetic modes simultaneously. In general, the suppression of forward scattering can be realized by the second Kerker condition. Here, a novel mechanism has to be adopted to explain zero-forward scattering, which originates from the complex interactions between dipolar and quadrupolar modes. However, for lossy and lossless core-shell spherical nanoparticles, zero-forward scattering can never be achieved because the real parts of Mie expansion coefficients are always positive. By adding proper gain in dielectric shell, zero-forward scattering can be found at certain incident wavelengths, which means that all electric and magnetic responses in Mie scattering can be counteracted totally in the forward direction. In addition, if the absolute values of dipolar and quadrupolar terms are in the same order of magnitude, the local scattering minimum and maximum can be produced away from the forward and backward directions due to the interacting effect between the dipolar and quadrupolar terms. Furthermore, by adding suitable gain in shell, super-forward scattering can also be realized at certain incident wavelengths. We also demonstrated that anomalously weak scattering or superscattering could be obtained for the core-shell nanoparticles with suitable gain in shell. In particular, for such a choice of suitable gain in shell, we can obtain zero-forward scattering and anomalously weak scattering at the same wavelength as well as super-forward scattering at another wavelength. These features may provide new opportunities for cloaking, plasmonic lasers, optical antennas, and so on.